Definition df-rediv 42434

Description: Define division between real numbers. This operator saves ax-mulcom 11092 over df-div 11797 in certain situations. (Contributed by SN, 25-Nov-2025.)

Assertion

Ref	Expression
df-rediv	⊢ /ℝ = (𝑥 ∈ ℝ, 𝑦 ∈ (ℝ ∖ {0}) ↦ (℩𝑧 ∈ ℝ (𝑦 · 𝑧) = 𝑥))

Distinct variable group:   𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-rediv

Step	Hyp	Ref	Expression
1		crediv 42433	. 2 class /ℝ
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cr 11027	. . 3 class ℝ
5		cc0 11028	. . . . 5 class 0
6	5	csn 4579	. . . 4 class {0}
7	4, 6	cdif 3902	. . 3 class (ℝ ∖ {0})
8	3	cv 1539	. . . . . 6 class 𝑦
9		vz	. . . . . . 7 setvar 𝑧
10	9	cv 1539	. . . . . 6 class 𝑧
11		cmul 11033	. . . . . 6 class ·
12	8, 10, 11	co 7353	. . . . 5 class (𝑦 · 𝑧)
13	2	cv 1539	. . . . 5 class 𝑥
14	12, 13	wceq 1540	. . . 4 wff (𝑦 · 𝑧) = 𝑥
15	14, 9, 4	crio 7309	. . 3 class (℩𝑧 ∈ ℝ (𝑦 · 𝑧) = 𝑥)
16	2, 3, 4, 7, 15	cmpo 7355	. 2 class (𝑥 ∈ ℝ, 𝑦 ∈ (ℝ ∖ {0}) ↦ (℩𝑧 ∈ ℝ (𝑦 · 𝑧) = 𝑥))
17	1, 16	wceq 1540	1 wff /ℝ = (𝑥 ∈ ℝ, 𝑦 ∈ (ℝ ∖ {0}) ↦ (℩𝑧 ∈ ℝ (𝑦 · 𝑧) = 𝑥))

This definition is referenced by:  redivvald  42435